Competitive equilibrium with indivisible objects∗

We study an exchange economy with indivisible objects that may not be substitutes for each other, and we introduce the p-substitutability condition, a relaxation of the gross substitutes condition of Kelso and Crawford (1982), in which a parameter vector p is adopted to permit complicated types of complementarity. We prove that for any economy E, there exists a corresponding vector pE such that the pE-substitutability condition is sufficient to guarantee the existence of a competitive equilibrium, and that the largest competitive price of each object is equal to its contribution to the social welfare. Our analysis relies on a classification result which shows that the set of economies can be partitioned into disjoint similarity classes such that an economy has a competitive equilibrium whenever it is similar to another economy with an equilibrium.